Locally contractible metric spaces have singular constant sheaf cohomology

(Flasque sheaves are acyclic.) For an injective $k$-sheaf $I$, the monomorphism $j_{U!}{k}_U\to j_{V!}{k}_V$ for opens $U\subseteq V$ is the identity on stalks in $U$ and has zero source elsewhere. The adjunction [F10], restricted to $k$-linear maps, identifies Hom from these sheaves with $I(U)$ and $I(V)$: a map from $k_U$ is determined by its value on $1$, and locally constant scalar multiples are defined by gluing. Thus injectivity makes $I(V)\to I(U)$ onto, so $I$ is flasque. Embed a flasque $F$ into an injective $I$ using [F7], with quotient $Q$. By [F8], $I(U)\to Q(U)$ is onto and $Q$ is flasque. The long exact sequence and injective acyclicity [F11] give $H^1(X,F)=0$ and $H^{n+1}(X,F)\cong H^n(X,Q)$ for $n\ge 1$. Induction, uniformly over flasque sheaves, proves their acyclicity. The same proof on any open subspace gives the assertion there. Repeating the argument for abelian sheaves, with ${\mathbb{Z}}_U$ in place of $k_U$ and [F14] for the injective data, proves abelian flasque acyclicity as well. No $k$-injective is claimed to be injective among abelian sheaves.

(Global representatives.) Let $C^n(U;k)$ be the functions on singular $n$-simplices with image in $U$, with the usual cochain differential, and let ${\mathcal{C}}^n$ be its sheafification. Locally represent $s\in {\mathcal{C}}^n(U)$ by cochains on an open cover $\{U_j\}$; local representability follows from the stalk bijection [F6]. The open subspace $U$ is itself metric, hence paracompact and Hausdorff by [F1], [F2]. Apply [F3] in $U$ to obtain a locally finite open cover $\{W_i\}$ and assigned indices $j(i)$ with ${\overline{W_i}}^U\subseteq U_{j(i)}$, and write $c_i$ for the representative on $U_{j(i)}$. For each singular simplex in $X$ whose image lies in some $W_i$, choose one such index and assign the value $c_i(\sigma )$; assign zero when no such index exists. These choices use [A1] and define a global cochain $c\in C^n(X;k)$. The germ verification below shows that its sheafification restricts to $s$.

(Prescribed germs and flasqueness.) Fix $x\in U$. Local finiteness implies that $J_x=\{i:x\in {\overline{W_i}}^U\}$ is finite and some neighbourhood meets no $W_i$ for $i\mathrm{\notin }{J}_x$: first take a neighbourhood meeting only finitely many $W_i$, then avoid the closures of those not containing $x$. Closure containment from step 1.2 puts $x$ in every $U_{j(i)}$ for $i\in J_x$. The $c_i$ therefore have germs at $x$, all representing $s_x$. The stalk bijection [F6] makes their presheaf germs equal, so on one common smaller neighbourhood $N$ the finitely many cochains agree. Further shrink $N$ into a $W_{i_0}$ containing $x$ and away from all $W_i$ with $i\mathrm{\notin }{J}_x$. Every simplex in $N$ is then assigned the value of $c_{i_0}$, independently of its chosen admissible index. Thus $c$ represents $s$ at every point of $U$. Every section over $U$ has a global representative; restricting that representative to any larger open set gives an extension. Hence every ${\mathcal{C}}^n$ is flasque. The canonical sheafification map $C^n(V;k)\to {\mathcal{C}}^n(V)$ is also onto for every open $V$.

(The flasque resolution.) The augmentation $k_X\to {\mathcal{C}}^0$ sends a locally constant function to the same function on points. By [F6], [F13], the cohomology of its stalk complex at $x$ is the colimit of the singular cohomology of neighbourhoods of $x$. Contractible open neighbourhoods are cofinal. A contraction and the cochain dual of [F5] identify their cohomology with that of a point. The ordinary singular cochain complex of a point has $k$ in every nonnegative degree, with differential $k\to k$ equal to zero in even degrees and the identity in odd degrees, because the boundary is the alternating sum of $n+1$ identical faces. Its cohomology is consequently $k$ in degree zero and zero otherwise. The augmentation induces the degree-zero stalk isomorphism, and all positive stalk cohomology vanishes. Therefore $k_X\to {\mathcal{C}}^0\to {\mathcal{C}}^1\to \cdots$ is exact. By steps 1.1 and 2.1 its terms are acyclic, and [F12] applies: the injective datum [F7] is defined on every successive kernel. Its section complex computes the $k$-linear derived cohomology. The same resolution consists of acyclic underlying abelian sheaves by step 1.1; applying [F12] with [F14] identifies its section cohomology also with the published abelian-sheaf convention.

(Identification with singular cohomology.) The canonical sheafification map $C^{\ast }(X;k)\to \mathrm{\Gamma }(X,{\mathcal{C}}^{\ast })$ is surjective by step 2.1. Let $N^{\ast }$ be its kernel. A cochain lies in $N^{\ast }$ exactly when it vanishes on all simplices contained in some member of some open cover of $X$: a cochain has zero germ at $x$ exactly when it vanishes on all simplices of some neighbourhood of $x$ by [F6], and a cochain vanishing on the small simplices of a cover has zero germ everywhere, while conversely the neighbourhoods of all points form such a cover. For an open cover $\mathcal{U}$ let $K_{\mathcal{U}}^{\ast }$ be the subcomplex of cochains vanishing on the $\mathcal{U}$-small singular chains. By [F4] applied with coefficients in the field $k$, the inclusion of the small chain complex into all chains is a homology isomorphism, so the quotient chain complex is acyclic; over a field an acyclic chain complex of possibly infinite-dimensional spaces splits degreewise into its boundaries and a complementary subspace, and dualizing such a splitting shows that the Hom of the quotient into $k$ is acyclic. Since restriction of functions to the subcomplex of small chains is surjective, this dual complex is exactly $K_{\mathcal{U}}^{\ast }$; hence $K_{\mathcal{U}}^{\ast }$ is acyclic. The complexes $K_{\mathcal{U}}^{\ast }$ are directed by refinement, and their union is $N^{\ast }$. A cocycle in the union lies in one $K_{\mathcal{U}}^{\ast }$ and is a coboundary there, so $N^{\ast }$ is acyclic. The long exact sequence of the short exact sequence of cochain complexes $0\to N^{\ast }\to C^{\ast }(X;k)\to \mathrm{\Gamma }(X,{\mathcal{C}}^{\ast })\to 0$ therefore identifies $H_{\mathrm{sing}}^{}(X;k)$ with the cohomology of $\mathrm{\Gamma }(X,{\mathcal{C}}^{\ast })$.

(Comparison and naturality.) Steps 3.1 and 3.2 identify $H^q(X,k_X)$ with $H_{\mathrm{sing}}^{}(X;k)$ in all nonnegative degrees; both are zero in negative degrees. The comparison uses the canonical augmentation and sheafification map, not the choices used to prove surjectivity. For a continuous map $f:X\to Y$ between spaces satisfying the hypotheses, pullback of singular cochains on each open $V\subseteq Y$ gives cochains on $f^{-1}V$, commuting with differentials, restriction and the constant-sheaf augmentation. Sheafifying gives the same pullback morphism of resolutions. Thus the induced cohomology comparisons commute with pullback, proving naturality, in both the $k$-linear and abelian conventions compared in step 3.1.

(Open subspaces and finite-polyhedral consequences.) Every open subspace is metric and inherits the contractible-open basis, so the argument applies there. A nonempty contractible such space has the cohomology of a point by [F5]. For the remaining geometric claims, realize a finite simplicial complex in barycentric coordinates $(t_v)$ on its finite vertex set. If $x$ has support simplex $\sigma$, choose $0<\delta <{\min }_{v\in \sigma }{t}_v(x)$ and put $N_{\delta }(x)=\{y:\mathrm{\mid }{t}_v(y)-t_v(x)\mathrm{\mid }<\delta (v\in \sigma ),{\sum }_{v\mathrm{\notin }\sigma }{t}_v(y)<\delta \}$. These are open neighbourhoods and form a basis as $\delta \downarrow 0$. Every carrier simplex of $y\in N_{\delta }(x)$ contains $\sigma$, so the segment from $y$ to $x$ stays in that carrier and satisfies the defining inequalities; these segments contract $N_{\delta }(x)$ to $x$. The open star of a simplex $\rho$ is $\{y:t_v(y)>0(v\in \rho )\}$. It is star-shaped toward the barycentre of $\rho$, since each carrier contains $\rho$ and the segment preserves these positive coordinates. Its nonempty intersection with a closed simplex $\tau$ requires $\rho \subseteq \tau$ and has the same contraction inside $\tau$. The intersections of the neighbourhoods $N_{\delta }(x)$ with these subspaces form contractible open neighbourhood bases: their contractions toward $x$ preserve the star's positive coordinates and membership in $\tau$. All these spaces inherit a metric from the finite coordinate space. Therefore their constant-sheaf cohomology is $k$ in degree zero and zero otherwise. An open star need not be convex in the whole realization; only these explicit contractions are used.